Question

1. the graph below is a sinusoidal function. write a cosine and sine function with the smallest phase shift.

a - 6 marks

Explanation:

Step1: Determine the amplitude \(A\)

The amplitude \(A=\frac{\text{Max}-\text{Min}}{2}\). From the graph, \(\text{Max} = 7\), \(\text{Min}=- 3\). So \(A=\frac{7 - (-3)}{2}=\frac{10}{2}=5\)

Step2: Determine the vertical shift \(D\)

The vertical shift \(D=\frac{\text{Max}+\text{Min}}{2}\). So \(D=\frac{7+(-3)}{2}=\frac{4}{2} = 2\)

Step3: Determine the period \(T\)

The period \(T = 2\pi\). Using the formula \(T=\frac{2\pi}{B}\), since \(T = 2\pi\), then \(B = 1\)

Step4: Write the cosine function

The general form of a cosine function is \(y=A\cos(B(x - C))+D\). For the cosine function, if we assume the phase - shift \(C\) is the smallest. The standard cosine function \(y = \cos(x)\) has a maximum at \(x = 0\). Our function has a maximum at \(x=0\). So the cosine function is \(y = 5\cos(x)+2\)

Step5: Write the sine function

The general form of a sine function is \(y=A\sin(B(x - C))+D\). The standard sine function \(y=\sin(x)\) has a maximum at \(x=\frac{\pi}{2}\). We want to shift the sine function. Using the identity \(\sin(x+\frac{\pi}{2})=\cos(x)\). So the sine function is \(y = 5\sin(x+\frac{\pi}{2})+2\)

Answer:

Cosine function: \(y = 5\cos(x)+2\); Sine function: \(y = 5\sin(x+\frac{\pi}{2})+2\)